Nuprl Lemma : extend-approx-ball_wf
∀[k,n:ℕ]. ∀[p:unit-ball-approx(n;k)]. ∀[z:{-k..k + 1-}].
extend-approx-ball(n;p;z) ∈ unit-ball-approx(n + 1;k) supposing (Σ((p i) * (p i) | i < n) + (z * z)) ≤ (k * k)
Proof
Definitions occuring in Statement :
extend-approx-ball: extend-approx-ball(n;p;z),
unit-ball-approx: unit-ball-approx(n;k),
sum: Σ(f[x] | x < k),
int_seg: {i..j-},
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
member: t ∈ T,
apply: f a,
multiply: n * m,
add: n + m,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
unit-ball-approx: unit-ball-approx(n;k),
extend-approx-ball: extend-approx-ball(n;p;z),
int_seg: {i..j-},
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
and: P ∧ Q,
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
bfalse: ff,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
less_than': less_than'(a;b),
true: True,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q
Lemmas referenced :
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
int_seg_properties,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-le,
istype-less_than,
int_seg_wf,
sum_wf,
itermAdd_wf,
int_term_value_add_lemma,
unit-ball-approx_wf,
istype-nat,
sum-unroll,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
istype-top,
subtract_wf,
subtype_base_sq,
int_subtype_base,
add-subtract-cancel,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
squash_wf,
true_wf,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
dependent_set_memberEquality_alt,
lambdaEquality_alt,
sqequalHypSubstitution,
setElimination,
thin,
rename,
because_Cache,
hypothesis,
extract_by_obid,
isectElimination,
inhabitedIsType,
lambdaFormation_alt,
unionElimination,
equalityElimination,
sqequalRule,
productElimination,
independent_isectElimination,
applyEquality,
independent_pairFormation,
imageElimination,
hypothesisEquality,
dependent_functionElimination,
natural_numberEquality,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
universeIsType,
productIsType,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
addEquality,
multiplyEquality,
axiomEquality,
isectIsTypeImplies,
minusEquality,
lessCases,
axiomSqEquality,
imageMemberEquality,
baseClosed,
instantiate,
cumulativity,
intEquality,
promote_hyp,
functionIsType
Latex:
\mforall{}[k,n:\mBbbN{}]. \mforall{}[p:unit-ball-approx(n;k)]. \mforall{}[z:\{-k..k + 1\msupminus{}\}].
extend-approx-ball(n;p;z) \mmember{} unit-ball-approx(n + 1;k)
supposing (\mSigma{}((p i) * (p i) | i < n) + (z * z)) \mleq{} (k * k)
Date html generated:
2019_10_30-AM-11_28_14
Last ObjectModification:
2019_06_28-PM-01_56_14
Theory : real!vectors
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